I am reading through Rudin's Principles of Mathematical Analysis and had a few related questions.
First, Rudin defines an open set, $E$, as a set such that every point is an interior point. A point is an interior point if there is a neighborhood, $N \subset E$ that contains the point. This neighborhood depends on a metric. So in this book (at least so far), open sets are only considered in metric spaces. But open sets need not depend on a metric.
Now Theorem 2.34 says that compact subsets of metric spaces are closed and Theorem 2.35 says that closed subsets of compact sets are compact. I understand and agree with both of these proofs, but was wondering a bit about Theorem 2.35. Obviously, in a metric space an open subset of a compact set is not compact by Theorem 2.34.
My question is: Under what conditions is an open subset of a compact set compact? I don't know much topology, but are there some easy examples, or is this not possible?